Download The Effect of a Low Viscosity Layer on

Geophys. J. R. astr. SOC.
(1967) 14, 353-370.
The Effect of a Low Viscosity Layer on Convection
in the Mantle
J. Weertrnan
Summary
In this paper we attempt to find a solution of the following problem. It
appears reasonable to expect that if thermal convection occurs in the
Earth’s mantle, it may also occur within the Moon and Mars. The dimensions of these latter two bodies are comparable to the thickness of the
Earth’s mantle. Presumably the amount of radioactive heat generated
per unit mass is similar in all three bodies. Yet the surface morphology
of the Earth, which many scientists believe arises ultimately from mantle
convection, differs markedly from that of the Moon or Mars. The explanation advanced here for this difference is based on the effect produced on
convection in the mantle by the presence of a low ‘viscosity ’ or low creep
strength layer. It is assumed that the low velocity layer of the mantle is
such a low creep strength layer. A low viscosity layer changes the amount
of ‘coupling’ between the outer crust and mantle convection. (The crust
is ‘coupled’ if mantle convection produces stresses in the crust which are
large enough to deform it plastically. The crust is ‘decoupled’ from the
mantle if these stresses are insufficient to produce plastic deformation.)
The theory assumes that the viscosity or creep strength is essentially zero
in the low viscosity layer. The analysis is similar to that developed
earlier for the calculation of stresses within the mantle. We find that the
deeper within the mantle lies the low viscosity layer the greater is the
coupling of the outer crust to the mantle convection currents. If the low
creep strength layer lies close to the surface the outer crust is decoupled
from the interior. According to the literature the depth of the low
velocity layer is determined by the temperature and pressure profiles within
the mantle. If the temperature profile were kept constant but the pressure
reduced by changing g , the gravitational acceleration, the low velocity zone
would be moved to a level closer to the surface. It is proposed that because
of this interdependence between temperature, pressure and depth of the
low velocity layer, the low velocity layers in Mars and the Moon (if such
layers exist in these bodies) lie much closer to the surface than does the
corresponding layer in the Earth. Hence the outer crusts of Mars and the
Moon are more nearly decoupled from the interior than is the Earth’s crust.
Introduction
In this paper we attempt to find a solution of the following problem. It appears
reasonable to expect that if thermal convection occurs in the Earth’s mantle, it also
may occur within the Moon and Mars. The dimensions of these latter two bodies
are comparable to the thickness of the Earth’s mantle. Presumably the amount of
radioactive heat generated per unit mass is similar in all three bodies. Yet the surface
353
3 54
J. Weertman
I------
___-__-
I
,
X
-I
i
To + AT
FIG. I . Thermal convection of the mantle in the plane layer approximation.
morphology of the Earth, which many scientists believe arises ultimately from mantle
convection, differs markedly from that of the Moon or Mars. The surfaces of Mars
and the Moon present a serious obstacle to the acceptance of the importance, and
indeed the existence, of thermal convection currents within the Earth’s mantle.
We shall attempt to show in this paper that the differences between the surface
morphology of the Earth and of the Moon and Mars do not necessarily disprove the
existence of convection within all three bodies. In our theory we shall use an approximate ‘block’ model to analyse thermal convection. The basis of this analysis will
be described in the next section. We wish to make clear now that there is an important
difference between our treatment of convection in the mantle and that usually discussed in the literature (i.e. Rayleigh’s classical theory: Knopoff 1964 Appendices I,
11, 111, Heiskanen & Vening Meinesz 1958 p. 408, Chamalaun & Roberts 1962).
The conventional treatment of mantle convection makes use of an approximation
that linearizes the basic equations. Thus the physical situation treated by this approximation is a convective motion in which the major amount of net heat transport between
two surfaces occurs by conduction. The heat transported by convective motion is
only a minor perturbation of the total. When it is remembered that in the case of
mantle convection the major portion of the heat is transported by means of convective
motion and only a minor portion by conduction, it can be appreciated that the usual
treatment of convection in the mantle makes use of the worst possible limiting
approximation,
Basis of theory
Only two-dimensional problems will be considered in this paper. For simplicity,
mantle material will be treated as incompressible.*
Convection in the mantle is approximately similar to convection between two flat
rigid surfaces such as shown in Fig. 1. It is assumed that the rigid surfaces are lubricated
so that no shear stress is transmitted across them. The top surface is maintained at
the temperature To and the bottom surface at the higher temperature To AT. If no
convection occurs, the equilibrium temperature profile T is simply T = To A T(l -z/d)
where d is the thickness of the mantle and z is vertical distance. In this situation the
total heat transport between z =0 and z = d occurs by conduction.
Fig. 1 shows the particle motion for the fundamental mode of convective motion.
If the convective motion is very slow the temperature profile is perturbed only slightly
from the value just given. It is possible to obtain an almost exact solution of this problem if it is assumed that the mantle material is a linear viscous solid and that the
viscosity is constant and independent of depth and temperature (Knopoff 1964).
In Fig. 1 the top and bottom of the mantle are constrained to be planar surfaces.
A more realistic picture of mantle convection is shown in Fig. 2. The top and bottom
surfaces are free to assume any shape.
+
+
* A n analysis could be developed for compressible matter. In this situation the adiabatic
temperature profile must be considered.
The effect of a low viscosity layer
355
T
C
TENSION
FIG.2. Thermal convection of the mantle when the top and bottom surfaces are
free to assume any shape.
The velocities, strain rates, stresses, etc. for the situations pictured in Figs. 1 and 2
obviously are almost the same. These velocities, stresses, and strain rates, as well as
the temperature, are smoothly varying functions of z and the horizontal distance x.
The regions of compressive and tensile deformation are indicated in Fig. 2.
We make the basic assumption that the convection shown in Fig. 2 is approximated
to a reasonable accuracy by the situation shown in Fig. 3. In this figure the longitudinal
strain rate C: is tensile and constant throughout the dashed blocks I and 111; in blocks
I1 and IV the strain rate C: is compressive, constant, and equal in magnitude to the
tensile strain rate*. The shear strain rate ,:C is taken to be zero within all four blocks
(but not necessarily along their boundaries). It further is assumed that the horizontal
derivative of temperature i3T/ax is zero in all blocks. The boundaries of the blocks,
of course, represent singularities where strain rates, temperature, etc. change discontinuously. The model will be refined later in the paper by relaxing some of these
assumptions.
For the following reasons we feel that the ' block ' model of Fig. 3 is not too great
a departure from physical reality. (1) The low velocity solution of Fig. 1 (Knopoff
1964, Heiskanen &Vening Meinsez 1958) shows that velocities, strain rates, etc. depend
sinusoidally on x and z with a wavelength d in the vertical direction and A in the horizontal direction, where L=2J2d. Fig. 3 represents a square wave dependence on
these quantities, which is a rough approximation to a sinusoidal solution. (2) The
critical Rayleigh number for Fig. 3, which will be obtained shortly, is about the same
as that of Fig. 1. (3) If the (deviator) stresses acting in the mantle are larger than
the critical resolved shear stress of mantle rock (which at high temperatures presumably
. -
___
i-
-~
~
0
TO
0'
0
To+ AT
LIQUID CORE:
0'
p,=Zp
FIG. 3. Block model of convection that approximates Fig. 2.
* Throughout this paper the strain rates and stresses which appear in the equations are considered
to be positive quantities whether they are tensile or compressive in character. The nature of the
various stresses and strain rates will be obvious from the context.
356
J. Weerhan
is of the order of I to 10 bars) plastic deformation will take place by dislocation motion.
The high temperature steady-state creep rate i resulting from dislocation motion in
crystalline material usually is of the form (Kennedy 1963, Garofalo 1965, Weertman &
Weertrnan 1965)
B = Bun.
(1)
Here n is a constant (nz 3 to 5) and B = B, exp (- Q/kT),where Bo is another constant
and Q is a diffusion activation energy. The activation energy Q is pressure dependent.
(If the stresses in the mantle fall below the criticial resolved shear stress creep deformation arising from dislocation motion should be relatively unimportant. Gordon
(1965) has shown that for this situation Herring-Nabarro diffusional creep is likely
to control the creep flow in the mantle. The creep rate of diffusional creep is
i = Au = u/q,
(2)
where A = A , exp ( - Q / k T). The quantity A , is a constant, Q is the same, or approximately the same, activation energy mentioned before, and q is the coefficient of
viscosity.) Equation (1) predicts a much more rapid increase of creep rate with stress
than is found for a perfectly viscous solid whose creep rate is given by equation (2).
In fact equation (1) approximates the behaviour of a perfectly plastic solid, that is, a
material for which n approaches infinity. Plastic deformation occurs in a perfectly
plastic solid at a constant stress level. Since blocks I-IV are being deformed at a
constant stress level, their physical state is similar to that described by equation (1).
Calculation of stresses of Fig. 3
Blocks I and I1 of Fig. 3 are in upwelling convective motion and have higher than
average temperatures. The reverse holds true for blocks 111and IV. I f p is the average
density of the mantle, the average density of blocks I and I1 is p - Ap and of blocks 111
and IV is p Ap, where 2Ap/p = uA6. Here c( is the coefficient of thermal expansion
and A6 is the average temperature difference between blocks I and I1 and 111 and IV.
The mantle of model Fig. 3 is floating on a liquid core. Since the density pt of the
liquid core is approximately twice that of the mantle, for convenience we shall set
p L = 2 p . For this value of pL the total thickness of blocks I and I1 is equal to that or
111 and IV but they are displaced vertically from each other by an amount Ad given by
+
Ad=d(Ap[p).
(3)
This equation is derived easily. Let the thickness of blocks I and 11 be set equal to
d‘, where d’ is not necessarily equal to d . In Fig. 3 the pressure P at a depth equal to
the level of the base of I11 must be the same for all values of x. Below blocks I11 and
IV the pressure P equals ( p Ap)gd, whereas below I and II it is
+
(p
- Ap)gd’ + A d p , = ( p - Ap)gd’ +2pAd.
The two pressures are the same if d ’ = d and Ad is given by equation (3).
It also must be shown that the total force exerted across section 00 of Fig. 3 above
the level equal to the base of block I11 is identical to the force across section 0’ 0‘.
If these forces were unequal, Fig. 3 would not represent an equilibrium situation in
which the dimensions of the blocks remain constant with time. If it is assumed that
the density in blocks I and I1 (or I11 and IV) is uniform and equal to the average density
within the two blocks, the total force acting across 0’0‘is simplyf(p + Ap)gd’; across
00 it is +(p- Ap)gd” +&ptg(Ad)‘ (p - Ap)gd‘ Ad. (It should be noted that by
symmetry the deviator stresses acting across 00 or 0’0‘cancel out since one block is
extending and its counterpart is compressing.) These forces equal each other if
d ‘ = d , Ad is given by equation (1) and terms in (Ad)’ are dropped.
+
The effect of a low viscosity layer
x-----------i
4-p
351
0’
5c--
FIG. 4. Stresses acting on blocks I and IV of Fig. 3.
The deviator stresses acting across the faces 00 and 0’0’can be estimated by
calculating the forces acting on blocks I and 1V. Consider Fig. 4. The total force acting
on blocks I and IV arises from the hydrostatic pressure, the tensile deviator stress a
causing extension of block I, the compressive deviator stress a causing compression in
block IV, and the shear stress axzexerted at the bottom surface. If the stresses within
the blocks of Figs. 3 and 4 were smoothly varying functions of position as they are in
the convection pictured in Figs. 1 and 2, it would be reasonable to expect that aXz%a.
Therefore let oxzz a. Further, assume that the wavelength A of Fig. 3 is approximately
2d. The total force (directed to the left) acting on blocks I and 1V arising from a
and a,, is approximately 3od/2. The force due to hydrostatic pressure P , on block I
is t ( p - Ap)gd whereas the total force due to pressures P , and P, is
(p
+ Ap)gd -3p d Ad.
The pressures P I , P , and P , are indicated in Fig. 4. Setting the sum of the forces
equal to zero results in
a Egd Ap/8 = p g d O! 68/16.
(4)
The creep rate in blocks I-IV is found by placing this stress into either equations
(1) or (2). (According to equations (1) and (2) the creep rate is both temperature
and pressure dependent. This dependence will be ignored throughout our paper.)
Calculation qf average temperature diflerence A0
The stress given by equation (4) depends on A p and hence on the average
temperature difference between the upwelling and sinking convective currents. This
temperature difference can be found from the equation of heat flow. If heat flow in the
horizontal direction is neglected the equation of heat conduction in a moving medium
under steady state conditions (aT/at=O) reduces to the following:
ti(d2 T/dz2)- wd T/dz=0 ,
(5)
where K is the thermal diffusivity and tv is the velocity of the medium in the vertical
direction. If the origin (z=O) is situated at the bottom of block I1 for the upwelling
case or at the bottom of block I11 for the sinking case velocity w is given by: w=8z
in block 11; w = S ( d - z ) in block I; w = -8z in block 111; and w = -B(d-z) in block
IV. The creep rate in these equations always is considered to be a positive quantity.
The solution of equation (5) subject to the boundary condition that T = To+ A T at
z=O andT= To at z = d is
2
XU= T o + A T - ( A T / y ) / exp(-k2/2ti)dz
0
(6a)
358
J. Wecrtman
in block I11 where
di 2
y=
1 [txp(-iz2/2ic)+exp(-i(d2-z2)/2~)]dz
0
and
*/2
TIV= To+AT-(AT/y)
exp(-iz2/2rc)dz
0
2
- (AT/?) exp ( -id2/4rc)
exp ( i ( d - - ~ ) ~ / d2z~ ) (6c)
dl2
in block IV. The temperature TIIin block I1 is found by substituting -i for i in equations (6a) and (6b); similarly the temperature TI is found by making the same substitution in equations (6c) and (6b).
For small creep rates these equations reduce to
e r r = (GA Tz/K~)(
[d2/8]- [z2/6])= -om,
J
OI= (SAT/icd)(d-z)([d2/8]- [ ( d - ~ ) ~ / 6=
] -ON,
where the temperature T= To+AT(l-z/d)+O
temperature.
The average temperature difference A0 is
(7a)
(7b)
and 0 is a perturbation on the
AO= 5iA Td2/96~
(8)
when the creep rate is small. The stress resulting from this temperature difference is
found by inserting equation (8) into equation (4). If it is assumed that the mantle is
a perfectly viscous solid equation (4) in turn may be inserted into equation (2). The
resultant equation is
R = p g d 3 a AT/qrc=96~16/5=300,
(9)
where R is the dimensionless Rayleigh number.
In the (almost) exact solution of convection given in Fig. 1 it is found (Knopoff
1964) that the Rayleigh number equals 657. Our very crude analysis thus leads to a
not unreasonable estimate of the Rayleigh number. (The two values could be brought
into closer agreement by taking R=4d instead of 2d. Also it should be realized that
the exact solution of Fig. 2 in which there are no constraints to vertical displacement
at the upper and lower surfaces probably would result in a Rayleigh number somewhat
lower than 657.) This result gives us confidence that the crude block model of convection is physically plausible.
If the creep rate is large the temperature profiles TI, Tit, etc. in blocks I, 11, etc.,
reduce to
TIIZ 7 i To+AT
~
(104
for all values of z in the range OGzSd-z’, where z ’ x ( 2 ~ / i ) ~for
; d-z’,<z<d
TIx To+ (z* AT/y)(i/ic)f(1-iz*2/6rc+i2z*4/40rc2~,
(lob)
where z* =d -2;
TIII 2 TIVr Ta
(10c)
The effect of a low viscosity layer
359
for z ’ S z 6 d ; and
Xri
+
+
To A T - (zA T/y)(i/~)*(
1 - i z 2 / 6 ~ 3’ z ~ / ~ O K ~ ) .
(1 0 4
In all these equations y ~ 3 ( 2 n ~ / 8 ) * + + ( 2 ~ /-d(1i )/ 4 d ) ( 2 ~ / d 8 ) ~ .
It can be seen that the average temperature difference now is
AtIzAT.
(1 1)
It is virtually independent of the strain rate. The creep rate ic above which equation
(1 I ) is valid and below which equation (8) is valid is approximately
3, M
(12)
2K/d2.
For creep rates below this critical value the major portion of heat transport is by
conduction. At creep rates above 3, convective heat transport predominates.
The stress arising from the temperature difference may be found by substituting
equations (8) and (11) into equation (4). Let this ‘thermal’ stress be labelled crT.
Fig. 5 shows schematically the dependence of crT on creep rate for various values of
AX Also shown in this figure is the relationship between stress and creep rate given
by the power law of equation (1). This stress is labelled 6,. (The stress (cr$ given by
the viscous creep equation (2) also is plotted.) The points at which the curves cross
or are tangent to each other represent physically possible convective states. At these
intersections or tangencies the stress required to maintain a given creep rate is equal
to the thermal stress arising from the creep rate. Points B and C of Fig. 5 represent
states of stable steady-state convective motion. If the creep rates were increased at
either of these points the stress required to drive convection would be greater than
the thermal stress available. If the creep rate were reduced an excess thermal stress
would be available to speed up the convective motion. By the same argument point A
represents unstable steady-state convection. The points at which the curve of cr,,
is tangent to crT (for 3 < 8,) correspond to a condition of marginal (or neutral) stability
CONDUCTIVE HEAT
CONVECTIVE HEAT TRANSFER
-
c
ln
v)
W
tY
ln
c
FIG.5. Stress versus creep rate relationships for power law creep equation (curve
u J ; viscous creep (on); thermal stress u T i ; and for constant heat transport on.
3 60
J. Weertman
(Knopoff 1964). The intersections to the right of iC (i2 ic) represent convective flow
in the mantle.
Under steady-state conditions the heat transported per unit time between the
surfaces z=O and z=d must equal the total heat generated per unit time. Radioactive
sources of heat are distributed throughout the mantle and, of course, this fact should
have been taken into account in equation (5). We shall assume for simplicity that
the heat source resides at the core-mantle boundary rather than be distributed
throughout the mantle.
Fig. 5 is altered as follows when convection is in a higher than the fundamental
mode. Let M = 1, 2, 3, etc. represent the mode number. The fundamental
mode is M = 1. The curves of on and CT,, are unaltered by increasing M [i.e.
( r ~ , )=~ ( r ~ , ) , = ~ and (aJM= (c,JM=
l].
The critical creep rate L, is increased
by a factor of M 2 [i.e., (8c)nf=M2(8c)M=1]. The initial slope ~ , ‘ = d a ~ / dofi the
thermal stress curve in the region to the left of 8, is reduced by a factor M 4 . The same
reduction occurs for the initial slope of the stress oH associated with constant heat
transport. Thus (c7Tr)M=M-4(+r)M=
and ( o ~ ~= M
’ -) 4~( o H ’ ) M , In the region
to the right of 8, the stresses (T, and oHare reduced by a factor of M 2 [i.e., (a,),=
M - ‘ ( C ~ ) and
~ = ~(o& = M - 2 ( ~ H ) M = 1 Suppose
].
convection is the major heat
transport mechanism. By combining equations (14) (after the right-hand side is
divided by M2),
(15) or (16) one can obtain the relationship AT = M2(8/pdc)(2Hq/org)*
for a viscous solid and
A T= M z ( 8 H / p d c ) ( 2 ~ / ~ g H > ” ” B
” +- ”(”
+
‘1
for a solid obeying creep equation (1). In these expressions H is heat transported.
If H is held constant and the convecting system is free to adjust the value of AT the
most likely value of M is 1. In the fundamental mode the tcrnperature difference is
minimized. It is to be expected therefore that convection within the Earth takes place
in the fundamental mode. On the other hand, if both H and A T are held constant, in
general, M will be greater than I in a convecting system. If conduction is the major
heat transport mechanism A T is not altered appreciably by convection. For a viscous
and C T curves
~ ~
are
solid M will assume that value for which the initial slopes of the CT,
almost the same as the slope of the a,,curve. Steady-state convection cannot occur
if the a, or the ancurve lies entirely above the ( c T ~= ) curve
~
of Fig. 5.
Refinement of the model
There is one very undesirable feature of the convective mode1 of Figs. 3 and 4.
In this model the temperature of the material moving from block I into block IV or
from block I11 into block I1 must change instantaneously. This can only happen if
heat is added or removed through some external agency at the boundary. For the
situation inwhich the major amount of heat is transported by conduction this physically
unrealistic feature of the model introduces no serious error. However, when heat is
transported primarily by convection the error is serious. (In fact, the model is a
perpetual motion machine!) We wish to show how the model may be refined to make
it physically realistic. To do so we shall have to take into account temperature variations in the horizontal direction.
Assume that the Earth at a given instant in time is in the state pictured in Figs. 3
and 4. Let the creep rate B be much larger than the critical value ic. The characteristic
distance (2lc/8)* thus is much smaller than the dimensions of the mantle. Now assume
that there is no temperature discontinuity between blocks I and IV (and between 1I
and 111). No heat is to be supplied or taken away a t these boundaries. The initial
temperature distribution will be that shown in Fig. 6(a). The shaded area indicates
regions where there is an appreciably temperature gradient. The thickness of these
regions is of the order of (2~/8)*.
361
The effect of a low viscosity layer
i--------
-10’
0
SHEAR
TENSION--
I
COMPRESSION-.-
P
L
-
Ip
SHEAR
4
II
TI
--COMPRESSION
~
TENSION
m
FIG. 6. Block model of convectionwhen convective motion is the major mechanism
of heat transport. (a) Temperature distribution of Fig. 3. (b) Steady-state
temperaturedistribution. (c) Block model and stresseswithin blockscorresponding
to (b).
Because so little heat can be transported by conduction, after a time the boundary
separating regions of temperature To AT and To must take the steady-state form
shown in Fig. 6(b). The temperature gradient is essentially zero everywhere except in
the shaded zone whose thickness is of the order of (2~/i)*.
Fig. 6(c) shows a more realistic ‘ block ’ model for convection when conduction is
not the major heat transfer mechanism. This model is based on Fig. 6(b). Again we
have blocks I and 111 in which tensile flow occurs and blocks I1 and IV in which compressive flow occurs. These blocks are separated by two ‘buffer’ blocks V and VI
in which there is neither compression nor tension but only a shear deformation. The
dimensions of these buffer blocks are comparable to those of the other blocks. The
calculation of the tensile and compressive deviator stresses in blocks I-IV is identical
to that done for Figs. 3 and 4 and gives identical results. The refined model leads to
essentially the same results as the cruder model.
+
Heat transport
We now are in a position to calculate the heat transported by a convective cycle
under steady-state conditions. Let H represent the heat generated per unit time and
per unit horizontal distance at the bottom of the mantle.
When 6 < i C almost all of the heat is transported by conduction. Therefore
H = K ATld,
(13)
where K is the thermal conductivity.
When i>& almost all the heat is transported by convective motion. The amount
transported can be estimated with the aids of Figs. 6(b) and 6(c). The net amount of
362
J. Weertman
heat transported upwards through the 1-11 and the 111-IV interfaces by vertical motion
of matter is approximately pcBdATL/8, where c is the specific heat. Since block V
is a t a higher temperature than block VI there will be heat conducted downward across
their interface. Heat will be conducted (there is no vertical motion of matter across
this interface) by the temperature gradient existing at the V-VI interface. This gradient
can be expected to be of the order of the gradient at the top surfaces of blocks I and V
or the bottom surfaces of blocks 111 and VI. Hence the amount of heat conducted
downward across the interface V-VI will be of the order of HA12 under steady-state
conditions. The total amount of heat transported thus is [(pcddATR/8) (HR/2)].
This quantity must equal the total supply of heat HA/2 at the bottom surface. Hence
-
H=pcBdAT/8.
(14)
(When i: is very much larger than dc H i s given by the equation in ' further comments '
at end of this paper rather than by equation (14).)
When the heat supply is constant and is limited, heat transfer by conduction or
convection will adjust the temperature difference AT existing between the top and
bottom surfaces to that value which is such that the amount of heat transported just
balances the amount created. For example, if the convection motion speeds up more
heat is transported and AT will decrease. The steady-state convective motion that
ultimately i s established is that for which ATand the creep rate d satisfy equations (13)
or (14). These values of d and ATalso must be compatible with equations (8) or (1 1)
and equations (1) and (2) as well as equation (4).
Consider Fig. 5. The dashed curve in this figure represents a thermal stress aT
corresponding to values of B and A T which lead to a constant value of H in equations
(1 3) and (14). Let this stress be labelled aH. The steady-state convective state is that
state for which the curves of uH,or, and a, (or a,,) all intersect at the same point. By
combining equations (I), (4), (11) and (14) the following equation is found for the
case of d>&:
06 = B(d/B)'/" = 0rgH/2C.
When equation (2) describes the creep flow of mantle material this equation is replaced
by
i: = (0cgH/2?c)*.
(16)
It is interesting to note that the thickness d of the mantle has dropped out of these
equations.
Equations (15) and (16) give reasonable values for B and d if it is assumed that H
equals the geothermal heat escaping at the Earth's surface (H x 4 0 cal/cm2/year).
If we take 01=2 x 10-5/"Cand c=5 x 1O7erg/cm3/deg (Knopoff 1964) and assume
that g is a constant throughout the mantle ( g z lo3cgs), we find that the product a8
is approximately equal to 0.4dyn/cm2/year. This product is equivalent to a 40 bar
stress and a creep rate of IO-'/year. These are reasonable values. A viscosity" of
cgs (which is in the range usually quoted for the Earth) substituted in equation (16)
predicts a creep rate of lO-'/year.
If equations (15) and (16) are applied to the Moon and Mars it is seen that, other
factors being equal, the creep rate B is reduced because g is smaller for these bodies,
A t the surface of the Moon g is a factor of 6 smaller than at the surface of the Earth
and in the case of Mars g is a factor of 10/4 smaller. Smaller values of g make it less
* MacDonald (1963) has pointed out that the viscosity of the Earth (as estimated from the time
cgs) that convection cannot occur in the mantle. This
lag of the Earth's flattening) is so large
objection can be overcome with the use of the power law creep equation (I). It gives an ' apparent '
viscosity equal to B-I dn-').
Thus the apparent viscosity estimated from different physical effects
such as the rate of flattening will depend on the stress level involved. The smaller the stress level
the greater will be the apparent viscosity measured.
The effect of a low viscosity layer
363
I
I
I
I
Y
I
L L O W STRENGTH
I
I
Ip
lu
ZONE
m
likely that appreciable convection occurs provided that the creep strengths in all three
bodies are comparable. However, since creep strength depends on both pressure and
temperature the creep strengths of the Moon and Mars will be lower than that of the
Earth if the temperature in all three are similar. The hydrostatic pressure within the
Moon and Mars is less than within the Earth.
Effect of a low strength layer
It has been postulated in the past that the low velocity layer of seismology is a
region of low strength (Anderson 1962). This layer occurs at a depth of the order of
100-200 km. The internal friction or damping of the mantle is much higher in this
region than deeper in the mantle. The rate of earthquake energy release seems to go
through a minimum here.
Gordon (1965) has shown very clearly how it is possible for the creep strength of
the mantle material to go through a minimum as the depth increases. The increase
in temperature with depth causes a decrease in creep strength. However, the increase
in pressure with depth causes an increase in the creep strength. A suitable pressuretemperature profile permits the existence of a region with very low creep strength.
Gordon’s analysis was applied under the assumption that Herring-Nabarro
diffusional creep is the dominant flow mechanism in the mantle. His analysis applied
to creep controlled by dislocation motion leads to the identical prediction of a low
creep strength layer. This similarity arises from the fact that both Herring-Nabarro
creep and the movement of dislocations at relatively high temperatures are controlled
by the same mechanism, the diffusion of atoms. The effect of temperature and pressure
is identical for both types of creep.
If the low strength layer occurs at the centre of the mantle (z=+d) it is easy to
predict its effect on convective flow in Figs. 3 and 4. In this case the shear stress
corresponding to ox, of Fig. 4 is zero. Thus the stress given by equation (4) will be
twice as big; the Rayleigh number of equation (9) will be halved; the creep rates
given by equations (15) and (16) will be increased; and the temperature difference
ATrequired to maintain convection will be decreased. In other words, it is easier to
establish convection.
Suppose the low strength layer is not a t the midpoint but is situated much closer
to the upper surface.* This situation is shown in Fig. 7. Two possible fundamental
modes of convection can be imagined for this assymetric situation. The deformation
may proceed in a manner similar to that given in Figs. 3 or 6. Again ‘blocks ’ must
be considered, but they are not of the same thickness. Tensile flow occurs in blocks I
and 111 and compressive flow in I1 and IV (see Fig. 7). The low strength zone forms a
*Throughout this paper the upper mantle is defined to be that region of the mantle lying
above the low strength layer.
24
364
J. Weertman
boundary between the upper and lower blocks. The situation pictured in Fig. 7
certainly would occur if the low strength zone were at the midpoint z=+d. If the low
strength zone were placed slightly above the midpoint ( z >+d) the physical state still
would be described by Fig. 7. However, it might be expected that the model would
break down if the low strength zone were placed close enough to the upper surface.
The low strength zone will be considered to be so weak that it is essentially a fluid.
The hydrostatic pressure in this region is a constant for all values of .x. This fact
determines the ratio DrlDrv, where Di and Dw are the thicknesses of the different
blocks.
Let p - Ap’ be the average density in block I; p Ap‘ the average density in block
1V; and p be the average density throughout the mantle. Similarly, let p - A p be
the average density in block I1 and p + Ap the average density in block 111.
The thicknesses Dr and DIV must satisfy the relationship
+
in order to ensure that the pressure is constant within the low strength zone. (Second
order terms have been dropped in this expression.) The tensile deviator stress CJI
acting across block I will have the same magnitude as the compressive deviator stress
acting across 11. From a balance of forces argument it is found that
The thickness of blocks 11 and 111 may be obtained from similar arguments:
Again second order terms have been dropped and it is assumed that pL equals 2p.
The deviator stresses in both blocks have the same magnitude and are equal to
a 1 1 =$gApDIrr.
(20)
Under steady-state conditions the vertical velocity is zero at the lower and upper
surfaces of the mantle. Hence the vertical velocity at the lower boundary of block I
is EI D1 and the velocity at the upper boundary of block 11is EII Drr. These two velocities
need not match each other. Mass balance can be maintained by flow of matter within
the low strengthzone, which has the properties of a fluid of low viscosity. (However,
unlike a fluid, the low velocity zone cannot be ‘squeezed out’ and thus eliminated by
the flow of matter in it. As long as the flow does not change the temperature-pressure
conditions unduly, the low strength zone will always regenerate itself.)
The flow of matter within the low strength layer will change the manner of heat
transport. The heat transport equation ( 5 ) has to be solved taking into account this
new boundary condition. To make the problem manageable we shall assume that
per unit time a uniform layer of mantle material of thickness (811 DII - i r 0 1 ) is removed
between blocks I and I1 and inserted between blocks I11 and IV. (Of course, the reverse
flow also is possible. It will become evident later that this situation is unlikely to
occur.) Material is removed from a region at a higher temperature to another region
a t a somewhat lower temperature. The low strength layer thus is a heat sink between
I and I1 and a heat source between 111 and IV. Any change in the heat budget between
blocks V and VI will be ignored.
Again let the temperature of the upper surface of the mantle be maintained at To
and the lower surface at To+ AT. Let the temperature between blocks I and 11 be fi
and between I11 and IV be TIV. Consider now the temperature profile found from
equation (5) for two limiting cases:
The effect of a low viscosity layer
365
Large creep rates in both upper and lorver mantle
Suppose that &1>2ti/D11~
and that BI>~K/DI'. According to equation (12) we
are in the region of high creep rates. Hence the temperature profile must resemble
the profile shown in Fig. 8(a). In the rising convective current the temperature remains
at T= To A Tto within a distance ZI'M (2ti/Q1)*of the upper surface. The temperature
Ti thus is To+ AT. In the descending current the temperature in block I V remains
at T= To to within a distance ZI' from the low velocity layer, at which point the temperature rises to T=TIv. The temperature in block 111 remains at this value to a
distance ZIII' % (2ti/E111)* from the bottom surface, at which point the temperature rises
to To AX The temperature TIVremains to be determined.
+
+
FIG. 8. Temperature distribution of the block model of Fig. 7. (a) Large
creep rates above and below the low strength layer. (b) Large creep rate below
the low strength layer and small creep rate above it.
The average difference in temperature between blocks I and IV is simply AT.
Equation (18) thus becomes
Similarly the stress CTIIIis given by
where A T f = To+ AT- TIV. Another expression for the stress can be found if the
creep rate BIII is known. This creep rate must satisfy the modified veision of equation
(14) in which BIII is substituted for 8 and DIII for 4 2 . Hence the stress CTIII is
if the creep law is the purely viscous relationship given by equation (2). Otherwise
the stress is
0111 = ( ~ H / ~ c DBAT')''".
III
(23b)
The following equations for the stress may be obtained from equations (22) and (23)
by eliminating the temperature difference AT':
366
J. Weertman
in the case of viscous creep and
(24b)
(TI11 = (2Hgr/cB)l/("+
')
for the other type of creep given by equation (1). An argument identical to that
used to obtain equation (24) can be invoked to obtain an expression for the stress (TI
in which the temperature difference A T does not appear. One finds that (TIZ(TIII.
Hence AT'/ATZDI/DII~. (If either the coefficient of viscosity q or the constant B
has a different value in the upper mantle than the lower mantle, the stress GI still is
given by equations (24) with (TI substituted for (TIII. However, the values of q and B
appropriate to the upper mantle must be placed in these equations. A higher creep
strength in the upper mantle requires that AT be larger.)
The stresses found by using equation (24) are approximately the same as those
given by equations (1 5) and (16). It should be noted that the temperature differences
AT' and ATrequired to obtain a stress of the order of 40 bars are not excessive. For
D1=200km, ATz66"C and AT'z5"C.
Small creep rates in tipper mantle and large creep rates in lower mantle
If the thickness of the upper blocks of Fig. 7 is made small enough, conduction
becomes the predominant mechanism for the transport of heat through the blocks.
Suppose that the strain rate in the lower mantle is high enough that convection is the
predominant heat transport mechanism there. The resultant temperature profile is
shown in Fig. 8(a). The same arguments used in the previous section lead to the conclusion that the average temperature difference AT' in the lower mantle is the same
as that previously calculated. The stress (TIII also is the same. The average temperature difference in the upper mantle is only about +AT'. Therefore the stress (TI is
small. (It is given by equation (21) with AT' substituted for AT.) One would not
expect the upper mantle to suffer appreciable creep deformation.
The temperature difference A T is given by equation (13) with DI substituted for
d in that equation. The criticial value of DI at which heat transfer mechanism in
the upper mantle changes from convection to conduction can be found from the
characteristic diffusion distance (i./2~)*. If i. again is taken to be 10-8/year and
lc=3 x 10-2cm2/s (Knopoff 1964) this characteristic distance is 140km. Thus if
the low strength layer were at a much shallower depth the upper mantle would not
deform plastically.
It might be argued that in the presence of a low velocity layer the convection loop
would not be as shown in Fig. 7 but rather would more closely resemble the model
shown in Fig. 9. This latter model is an approximation of Fig. 6. In Fig. 9 blocks I1
(compression), VI (shear), and 111 (tension) correspond to and have essentially the
same dimensions as the analogous blocks of Fig. 6(c). Blocks I' (tension), V' (shear
0'
0
I
1
I
FIG.9. Block model corresponding to Fig. 6(c) when convection currents
approximate those shown in Fig. 2 and there is little mass transport through the
the low strength layer
The effect of a Iow viscosity layer
367
and IVI (compression) likewise correspond to blocks I, V, and IV of Fig. 6(c). The
upper mantle blocks I (tension), V (shear), and IV (compression) are merely perturbation on the overall convective flow pattern which is, essentially, the same as Fig. 6(c).
If the heat transport through the upper mantle were by conduction the temperature
profile would be similar whether Figs. 9 or 7 described the convection in the lower
mantle. However, if the major amount of heat transport in the upper mantle were
by convection, Fig. 9 would represent an unlikely convection model. As we have
just learned in order that convection occur in the upper mantle the average temperature difference there ( A T ) must be a factor Dn/Dr larger than the average temperature
difference (AT’) in the lower mantle. This much greater temperature difference arises
from the transport of matter in the convective loop through the low strength layer.
If this transport is minimized, as is the case in Fig. 9, it would be impossible to establish
that AT> AT‘. (However, if in Fig. 9 block I were in compression and block 11 in
tension, the average temperature difference in the upper mantle would be a factor
DII/DJlarger than in the lower mantle. In this situation the mass transport of matter
in the low strength zone would be greater than in the model pictured in Fig. 7.)
Discussion
The results of this paper show that convection can take place in the upper mantle
even though there may be an underlying low strength layer which, over long periods,
cannot transmit shear stresses. (The resultant creep deformation in the upper mantle
can lead to still higher stresses in the Earth’s crust in the manner described earlier
(Weertman 1962, 1963).)
If the thickness of the mantle above the low strength layer is small the stresses in
this region are small and the upper mantle will not deform plastically. This latter
result suggests a possible explanation for the lack of widespread deformation (other
than impact craters) of the crust of the Moon and Mars. If these bodies contain low
strength layers, and if these layers lie closer to the upper surface than the corresponding
layer in the Earth, the crusts of the Moon and Mars cannot be deformed plastically
by thermal convection currents.
Of course it could be denied that convection currents exist at all in Mars and the
Moon. After all, it is still open to question whether convection even exists within the
Earth’s mantle. However, suppose one is a proponent of continental drift and therefore regards the existence of convection currents within the Earth as being almost
certain. If convection currents exist within the Earth’s mantle then arguments can
be advanced to support the view that they may exist within the Moon and Mais. It
is likely that the amount of radioactive heat generated per unit mass is about the same
for Mars and the Moon as it is for the Earth. If the Earth has been in a steady-state
convective motion over the major portion of its history and the Moon and Mars have
not, obviously the amount of radioactive heat retained in the interior of these latter
bodies is much larger than that retained within the Earth. The internal temperatures
of Mars and the Moon would be much higher than the Earth’s internal temperature
provided that the internal temperatures of all three bodies were similar at the time of
their formation. The hydrostatic pressures within Mars and the Moon are smaller
than within the Earth. Thus even if the internal temperatures of the Moon and Mars
were somewhat lower than that of the Earth the viscosity or the creep strength of the
Moon and Mars still could be much lower than that of the Earth because the reduced
pressure permits their interiors to be closer to the melting point. (That is, the diffusion
rate of atoms within a solid and hence the viscosity or creep strength depends exponentially on the proximity of a solid to its melting point. The diffusion rate is proportional
to a factor exp (- CT,/ T), where C is a constant and T, is the melting temperature.
The melting temperature of a normal solid is raised by increasing the pressure.) All
these factors suggest that if the Earth is in a state of convection so should be the Moon
368
J. Weertman
and Mars. The surface morphology of the Moon and Mars thus may present a serious
objection to the postulation of convection within the Earth’s mantle.
This objection to convection within the Earth can be answered in two ways. It
could be postulated that the interior temperatures at the time of the formation of the
Earth were very much higher than the corresponding temperatures for the Moon
and Mars. Thus these latter bodies may be so much colder than the Earth that their
viscosities or creep strengths are too great to permit convection at the present time.
Another answer to the objection is the analysis of this paper. Convection within Mars
and the Moon need not lead to plastic deformation of their crusts if their low strength
zones lie at shallower depths than the corresponding layer in the Earth. A theory such
as Gordon’s (1965) could be used to explain why the low strength zones may exist at
shallower depths in the Moon and Mars.
Final comments
At large strain rates (898,) the heat transported by convection no longer is given
by equation (14). The velocity of matter in the vertical direction is zero at the upper
and lower surfaces of a convecting layer. Hence the heat transported by convection
must equal the heat transported by conduction down the temperature gradient that
exists at these surfaces. This temperature gradient is found from equations (6) and
is equal to A T ( Q K ) * . Thus H , the heat transported by convection, is equal to
H = K AT(8/21~)+.
(14a’)
In general, the heat transport given by this equation will not equal that given by
equation (14). The magnitude of the difference between them will increase the large
the 8.
The heat transported by convention also must equal an expression proportional
to the temperature difference AT between the rising and sinking convective currents,
the average velocity of a convective current (-id), and the heat capacity pc. Equation
(14) was obtained from such an analysis. However, in determining equation (14) it
was assumed that blocks I-VI in Fig. 6 all were comparable in size. This situation is
true only if the strain rate 8 is of the same magnitude as the critical strain rate ic. At
larger strain rates the horizontal dimensions of blocks I-IV will be much smaller than
the horizontal dimensions of the buffer blocks V and VI. Thus the effective width of
the rising and sinking convective currents decreases as 8 increases. Therefore they
become less efficient in transporting heat.
Let 0represent the width in the horizontal direction of block I divided by the width
of block V. As 8 increases fi will decrease. The heat H transported through convective motion of matter can be expected to be of the order of
H = fipcid AT/8.
(14b’)
This equation is simply equation (14) of the text modified by the insertion of the factor
fi. Since equations (14a’) and (14b’) must lead to the identical heat transport the
term fi can be found by setting these equations equal to each other.
When p approaches zero the temperatures in Fig. 6 are equal to To+ AT almost
everywhere in the top blocks and to To almost everywhere in the bottom blocks. In
other words, an almost stable situation is reached. Hot, light matter resides over cold,
dense matter. Clearly this situation tends to reduce the stresses driving the convective
motion. It is simple to show that equation (4), which gives the stress driving convection,
is modified into the following one
CT=
ppgda AT/16.
(4‘)
(This last equation is obtained using an analysis similar to that employed for Fig. 4.
One first analyzes the forces acting on blocks I, IV, and V of Fig. 6 . These blocks are
considered as one unit just as in Fig. 4 blocks I and IV were considered as a unit. Then
The effect of a low viscosity layer
369
one analyzes the forces acting on block V alone. Equation (4') is the result of the
analysis.)
For a viscous solid o = qi. If this expression is inserted into equation (4') and then
this equation is combined with equations (14a') and (14b') one finds that the heat
transport is given by
H x ( o r g K 4 / 8 ~ r l ~ Z(AT)4/3.
)'/3
(25)
Thus the heat transport is proportional to (AT)"J3. This dependence agrees with that
found in laboratory experiments on heat transport at fast rates of convection. (See,
for example, Chandrasekhar (1961), Hydrodynamic and Hydromagnetic Stahilitj.,
p. 68, Fig. 13, Clarendon Press, Oxford.)
When /I and AT are eliminated from equation (4') through the insertion of equations (14a') and (14b') one finds that the stress is equal to
o = (i/B)'/"= ( C @ ~ B / ~ C ) " (I ")
(1 5')
B =vi = (orqgH/2c>)
(16')
for a non-viscous solid and
for a viscous solid. Surprisingly, these equations give stresses (as well as strain rates)
that are identical to those determined from equations (15) and (16) of the text.
U.S. Army Cold Regions Research and Engineering Laboratory,
Hanover,
N z w Humpshire.
Department of Materials Science* and Department of Geology,
Northwestern University,
Etianston,
Illinois.
1966 June.
References
Anderson, D. L., 1962. The plastic layer of the Earth's mantle, Scient. Am., July,
pp. 52-59.
Chamalaun, T. & Roberts, P. H., 1962. The theory of convection in spherical
shells and its application to the problem of thermal convection in the Earth's
mantle, in Continental Drift, (ed. by S . K. Runcorn). Academic Press, New York.
Garofalo, F., 1965. Fundamentals of Creep and Creep-Rupture in Metals. Macmillan,
New York.
Gordon, R. B., 1965. Diffusion creep in the Earth's mantle, J. geophys. Res., 70,
2413-2418.
Kennedy, A. J., 1963. Processes of Creep and Fatigue of Metals. John Wiley,
New York.
Knopoff, L., 1964. The convection current hypothesis, Rev. Geophys., 2, 89-122.
MacDonald, G. J. F., 1963. The deep structure of continents, Reu. Geophys., 1,
587-665.
Heiskanen, W. A. & Vening Meinesz, F. A., 1958. The Earth and its Gravity Field.
McGraw-Hill, New York.
* Support is acknowledged to the Advanced Research Projects Agency of the Department of
Defense through the Northwestern Materials Research Center of that portion of the research carried
out at Northwestern.
370
J. Weertman
Weertman, J., 1962. Mechanism for continental drift, J. geophys. Res., 67, 1133-39.
Weertman, J., 1963. The thickness of continents, J. geophys. Res. 68,929-32.
Weertman, J. & Weertman, J. R., 1965. Mechanical properties, strongly temperature
dependent, in Physical Metallurgy (ed. by R. W. Cahn), pp. 793-819. NorthHolland, Amsterdam.